Department of Mathematical Sciences
Rutgers University, Camden, NJ
- Textbooks (optional):
- E. Suli and D. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003. ISBN-10: 052100794, ISBN-13: 978-0521007948
- A. Ralston and P. Rabinowitz, A
First Course in Numerical Analysis, Second Edition, Dover
Publications. ISBN-10: 048641454X, ISBN-13: 978-0486414546
- Class meetings:
Wednesdays 6:00PM - 8:50PM Online
- Instructor: Dr. Steve Alessandrini, sma@rutgers.edu
- Office hours: By appointment.
- Prerequisites:
- Calculus.
- Knowledge of C++. Tutorial, SimpleExample.cpp
- C++ Style Guide for Scientific Computing
- This course is suitable for graduate students and advanced undergraduates who have the prerequisites.
- It is preferred that this first semester course 645:571 is taken before the second semester course 645:572.
- Course description:
This
one-year sequence covers various numerical techniques for solving
mathematical
problems on a computer. The first semester 645:571 covers the
IEEE
internal representation of floating point numbers, interpolation, root
finding, numerical integration, numerical differentiation, and function
minimization. The second semester 645:572 covers numerical linear
algebra and the numerical solution of differential equations. The
material is presented so that topics build on one another and
applications
are given to illustrate the use of the techniques.
- Notes
and Examples:
(PDF Format)
- Course Information
- Chapter 1: Introduction
- Part
1: pp 1 - 14 (9/2/2020)
- Example demonstrating why not to use pow for square roots and squaring: PowTest.cpp
- Chapter 2: Interpolation
- Part
2: pp 15 - 25 (9/9/2020)
- Part
3: pp 26 - 39 (9/16/2020)
- MATLAB Examples
- RungeExample.m
(This is a modified version from the one demostrated during
class. All cases are now on the same plot for easier comparison)
- PWLinear.m
- Hermite.m
- Spline.m
- RungeExampleSplineEven.m
- RungeExampleSplineOdd.m
- ClampedSplineExample.m
- KeyframeAnimation.m
(This is a modified version from the one demostrated during
class. The distances are now more to scale)
- Cubic Spline Application: How fast could Usain Bolt have run?
- H.
K. Eriksen, J. R. Kristiansen, O. Langangen, and I. K. Wehus, "How fast
could Usain Bolt have run? A dynamical study," American Journal of
Physics, 77, 224-228 (2009).
- SmoothSpline.m
- CubicSplineExample.m
- LeastSquaresExample.m
- polyfitweighted.m
- Smoothing Spline References:
- C. H. Reinsch. (1967). "Smoothing by Spline Functions," Numerische Mathematik, 10: 177-183.
- C. H. Reinsch. (1971). "Smoothing by Spline Functions. II," Numerische Mathematik, 16: 451-454.
- Monotone Spline:
- References:
- F. N. Fritsch and R. E. Carlson. (1980). "Monotone Piecewise Cubic Interpolation". SIAM Journal on Numerical Analysis. 17 (2): 238–246.
- R. Dougherty, A. Edelman, and J. Hyman. (1989). "Nonnegativity-, Monotonicity-, or Convexity-Preserving Cubic and Quintic Hermite Interpolation". Mathematics of Computation. 52: 471-494.
- G. Wolberg and I. Alfy. (2002). "An energy-minimization framework for monotonic cubic spline interpolation". Journal of Computational and Applied Mathematics. 143(2): 145-188.
- MATLAB:
- Chapter 3: Roots of Equations and Optimization
- Part
4:
pp 40 - 48 (9/30/2020)
- Part
5:
pp 49 - 60 (10/7/2020)
- Part 6: pp 61- 70 (10/14/2020)
- MATLAB Examples
- NewtonSqrtExample.m
- PerformLineMinimization.m
- ComputeFunctionAndGradient.m
- CircleFitSteepestDescent.m
- CircleFitConjugateGradient.m
- CircleFitDFP.m
- CircleFitBFGS.m
- CircleFitLM.m
- CircleFit.m
- FindGlobalMinimumExample.m
- Chapter 4: Numerical Integration
- Part 7:
pp 71 - 77 (10/21/2020)
- Part 8:
pp 78 - 86 (10/28/2020)
- Part 9: pp 87 - 97 (11/4/2020)
- Part 10: pp 98-105 (11/11/2020)
- MATLAB Examples
- Midpoint.m
- Trapezoidal.m
- Simpson.m
- TestSplineQuadrature.m
- TrapezoidalPeriodicFunction.m
- DetermineRGraphically.m
- IntegralTest.m
- H. Yserentant, "A remark on the numerical computation of improper integrals," Computing, Vol. 30, N0. 2, (1983), pp. 179-183.
- Adaptive Quadrature
- W. Gander and W. Gautschi, "Adaptive Quadrature—Revisited", BIT Numerical Mathematics, 40, 84–101 (2000).
- TestAdaptiveSimpson.m
- AdaptiveSimpson.m
- Chapter 5: Numerical Differentiation
- Part 11: pp 106 - 112 (11/18/2020)
- Part 12: p. 113 (12/2/2020)
- W. Squire and G. Trapp, "Using Complex Variables to Estimate Derivatives of Real Functions," SIAM Review, Vol. 40, No. 1, (March 1998), pp. 110-112.
- K.-L.
Lai, J. L. Crassidis, Y. Cheng, and J. Kim, "New Complex-Step
Derivative Approximations with Applications to Second-Order Kalman
Filtering," 2005 AIAA Guidance,Navigation, and Control Conference and
Exhibit, San Francisco, CA, USA, 15-18 August 2005, pp, 1-17.
- MATLAB Examples
- Differentiation of Noisy Data
- I. Knowles and R. J. Renka, "Methods for Numerical Differentiation of Noisy Data," Electronic Journal of Differential Equations, Conference 21 (2014), pp. 235-246.
- J. J. Stickel, "Data smoothing and numerical differentiation by a regularization method," Computers and Chemical Engineering, 34 (2010), pp. 467-475.
- MATLAB
- Homework
Assignments: See Canvas
- Notes:
- First Class: Wednesday 09/02/2020
- The three 1-hour
exams will be given on Canvas from 6:00 pm - 7:00pm on the following
Wednesdays: 9/23/20, 10/21/20, 11/18/20
- No Class: Wednesday 11/25/2020 - Thanksgiving Break
- Last Regular Class: Wednesday 12/9/2020
- Final
Exam Period: Wednesday 12/16/2020 6pm-9pm
- Rutgers University Academic Integrity Policy